Consider The Formula Used For Any Confidence Interval

Consider The Formula Used For Any Confidence Interval And The Elements

Consider the formula used for any confidence interval and the elements included in that formula. What happens to the confidence interval if you (a) Increase the confidence level, (b) Increase the sample size, or (c) Increase the margin of error? Only consider one of these changes at a time. Explain your answer with words and by referencing the formula.

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Confidence intervals are essential statistical tools used to estimate the range within which a population parameter, such as a mean, is likely to lie with a certain level of confidence. The general formula for a confidence interval for a population mean when the population standard deviation is unknown is expressed as:

CI = μ̂ ± t_α/2 * (s / √n)

where μ̂; represents the sample mean, t_α/2; is the critical value from the t-distribution corresponding to the confidence level, s is the sample standard deviation, and n is the sample size. This formula encapsulates key elements: the point estimate (sample mean), the margin of error (which depends on the variability in the data, sample size, and critical value), and the confidence level itself, through the critical t-value.

Analyzing how changes to the confidence level, sample size, or margin of error affect the confidence interval requires understanding the mathematical relationships within this formula. Each component influences the width — the range of the interval — in different ways.

(a) Increasing the Confidence Level

When we increase the confidence level (for example, from 95% to 99%), the critical value t_α/2; increases because the higher the confidence level, the wider the interval needs to be to capture the true parameter with greater certainty. Since the t_α/2; appears as a multiplicative factor in the margin of error, a larger t_α/2; results in a larger margin of error, thereby expanding the confidence interval. This is because higher confidence levels require more conservative estimates to ensure the true parameter is contained within the interval more frequently.

Mathematically, as the confidence level increases, t_α/2; increases, leading to:

• Wider confidence interval
• Greater range estimates

(b) Increasing the Sample Size

Increasing the sample size (n) has a different impact. Since the standard error of the mean (s / √n;) appears in the formula, larger n values decrease the standard error. Consequently, as n increases, the entire margin of error term reduces because;

Margin of Error = t_α/2; * (s / √n)

which decreases as √n; increases. This leads to a narrower confidence interval, implying more precise estimates of the population parameter. Notably, the critical t-value may decrease with increasing degrees of freedom (larger n), further contributing to narrower intervals.

(c) Increasing the Margin of Error

The margin of error is a direct component of the confidence interval width. If we artificially increase the margin of error (for instance, by accepting a larger variability or a looser criterion), the overall width of the confidence interval increases proportionally. Since the margin of error = critical value * standard error, increasing the margin of error equates to raising this product, leading to wider bounds of the interval and less precise estimation.

Summary

In conclusion, increasing the confidence level enlarges the confidence interval because the critical value increases. Increasing the sample size reduces the confidence interval's width due to the decreased standard error. Increasing the margin of error directly expands the confidence interval's bounds, reducing precision but increasing the likelihood that the interval captures the true parameter. Understanding these relationships helps in designing studies with appropriate levels of confidence, precision, and resource allocation.

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